Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T19:15:55.707Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Existence of Asymptotic-lp Structures in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Adi Tcaciuc
Adi Tcaciuc*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1 e-mail: tcaciuc@math.ualberta.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that if a Banach space is saturated with infinite dimensional subspaces in which all “special” $n$-tuples of vectors are equivalent with constants independent of $n$-tuples and of $n$, then the space contains asymptotic-${{l}_{p}}$ subspaces for some $1\,\le \,p\,\le \,\infty $. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.

Keywords

46B2046B4046B03
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 4 , 01 December 2007 , pp. 619 - 631
Copyright
Copyright © Canadian Mathematical Society 2007

References

[ADKM] Argyros, S. A., Deliyanni, I., Kutzarova, D. N., and Manoussakis, A., Modified mixed Tsirelson spaces. J. Funct. Anal. 159(1998), no. 1, 43–109.Google Scholar
[CO] Casazza, P. G. and Odell, E., Tsirelson's space and minimal subspaces. Texas Functional Analysis Seminar 1982-1983. Longhorn Notes, Universiy of Texas, Austin, TX, pp. 61–72.Google Scholar
[DFKO] Dilworth, S. J., Ferenczi, V., Kutzarova, D. N., and Odell, E., A remark about strongly asymptotic lp spaces and minimality. J. London Math. Soc. 75(2007), no. 2, 409–419.Google Scholar
[FFKR] Figiel, T., Frankiewicz, R., Komorowski, R. A., and Ryll-Nardzewski, C., Selecting basic sequences in φ-stable Banach spaces. Studia Math. 159(2003), no. 3, 499–515.Google Scholar
[GM] Gowers, W. T. and Maurey, B., The unconditional basic sequence problem. J. Amer.Math. Soc. 6(1993), no. 4, 851–874.Google Scholar
[J] Johnson, W. B., A reflexive Banach space which is not sufficiently Euclidean. Studia Math. 55(1976), no. 2, 201–205.Google Scholar
[JKO] Junge, M., Kutzarova, D., and Odell, E., On asymptotically symmetric Banach spaces. Studia Math. 173(2006), 203–231.Google Scholar
[K] Krivine, J. L., Sous espaces de dimension finie des espaces de Banach réticulés. Ann. of Math. (2) 104(1976), no. 1, 273—295.Google Scholar
[LT] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces. I. Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin, 1977.Google Scholar
[M1] Maurey, B., A note on Gowers’ dichotomy theorem. In: Convex Geometric Analysis, Math. Sci. Res. Inst. Publ. 34, Cambridge University Press, 1999, pp. 149–157.Google Scholar
[M2] Maurey, B., Quelques progrès dans la compréhension de la dimension infinie. In: Espaces de Banach classiques et quantiques, SMF Journ. Annu., Société Mathématique de France, Paris, 1994.Google Scholar
[MT] Milman, V. D. and Tomczak-Jaegermann, N., Asymptotic lp spaces and bounded distorsions. Contemp. Math. 144(1993), 173–195.Google Scholar
[OS] Odell, E. and Schlumprecht, T., The distortion of Hilbert space. Geom. Funct. Anal. 3(1993), no. 2, 201–207.Google Scholar
[Pe]Pelczar, A. M., Remarks on Gowers’ dichotomy. In: Recent Progress in Functional Analysis. North-Holland Math. Stud. 189, North-Holland, Amsterdam, 2001, pp. 201–213.Google Scholar
[S] Schlumprecht, T., An arbitrarily distortable Banach space. Israel J. Math. 76(1991), no. 1–2, 81–95.Google Scholar
[T] Tsirelson, B. S., Not every Banach space contains lp or c0 , Funct. Anal. Appl. 8(1974), 138–141.Google Scholar
[Z] Zippin, M., On perfectly homogeneous bases in Banach spaces. Israel J.Math. 4(1966), 265–272.Google Scholar